Problem: Multiply the following complex numbers, marked as blue dots on the graph: $(2 e^{5\pi i / 6}) \cdot (4 e^{4\pi i / 3})$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $2 e^{5\pi i / 6}$ ) has angle $\frac{5}{6}\pi$ and radius $2$ The second number ( $4 e^{4\pi i / 3}$ ) has angle $\frac{4}{3}\pi$ and radius $4$ The radius of the result will be $2 \cdot 4$ , which is $8$ The sum of the angles is $\frac{5}{6}\pi + \frac{4}{3}\pi = \frac{13}{6}\pi$ The angle $\frac{13}{6}\pi$ is more than $2 \pi$ . A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $\frac{13}{6}\pi - 2 \pi = \frac{1}{6}\pi$ The radius of the result is $8$ and the angle of the result is $\frac{1}{6}\pi$.